Steady flow energy equation and Bernoulli Equation

For steady flow energy equation , Consider a control volume, in which mass ∆m1/∆t enters through inlet with parameters

U = internal energy

v = Velocity

V = specific volume ( V1 = V2 )

P = pressure

Similarly mass ∆m2/∆t exits through outlet.

Therefore mass concentrationin control volume is

(∆m/∆t) control volume = ∆m1/∆t – ∆m2/∆t

•(∆E/∆t) in = internal energy+ flow work + kinetic energy+ potential energy + heat

Therefore , (∆E/∆t) in = ∆m1/∆t { U1 + P1V1 + v12/2 + gZ1 } + ∆Q/∆t

At exit , (∆E/∆t)out = ∆m2/∆t { U2 + P2V2 + v2 2/2 + gZ2 } + ∆W/∆t

(E/∆t) control volume= (∆E/∆t)in – (∆E/∆t)out

(E/∆t) control volume= ∆m1/∆t { h1 + v12/2 + gZ1 } + ∆Q/∆t – ∆m2/∆t { h2 + v2 2/2 + gZ2} – ∆W/∆t

Assumptions :

• (∆m/∆t) control volume = 0

• (E/∆t) control volume= 0

Here , (∆m/∆t) control volume = ∆m1/∆t – ∆m2/∆t = 0

So , ∆m1/∆t = ∆m2/∆t = m

Similarly,

(∆E/∆t)in = (∆E/∆t)out

Therefore ,

m{ h1 + v12/2 + gZ1 } + ∆Q/∆t = m{ h2 + v2 2/2 + gZ2 } + ∆W/∆t

this is Steady flow energy equation or SFEE .

Bernoulli Equation :

m{ h1 + v12/2 + gZ1 } + ∆Q/∆t = m{ h2 + v2 2/2 + gZ2 } + ∆W/∆t

Assume ,

(a)No heat transfer

(b) No work transfer

(c)No change in internal energy

(d)No change in density ( p = constant)

Then ,

m{ h1 + v12/2 + gZ1 } + ∆Q/∆t = m{ h2 + v2 2/2 + gZ2 } + ∆W/∆t

and

U1 + P1V1 + v12/2 + gZ1 = U2 + P2V2 + v2 2/2 + gZ2

As , Specific Volume = 1/ Density

i.e., V = 1/ p

Therefore

P1/p + v1 2/2 + gZ1 = P2 /p+ v2 2/2+ gZ2

P1/pg + v1 2/2g + Z1 = P2 /pg+ v22/2g+ Z2

This is Bernoulli Equation

Conclusion :

Hence both Steady flow energy equation and Bernoulli Equation are based on energy conservation and Bernoulli Equation is limited form of steady flow energy equation .

This is also known as the first law of thermodynamics for open system.